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double refraction on this hypothesis, but have not yet arrived
at any results sufficiently decisive to be communicated.'
Riemann also believed that his new geometry would prove of
scientific importance, as is shown by the conclusion of his
memoir (Clifford's translation):
•Either therefore the reality which underlies space must form
a discrete manifold, or we must seek the ground of its metric
relations outside it, in binding forces which act upon it.
"The answer to these questions can only be got by starting
from the conception of phenomena which has hitherto been
justified by experience, and which Newton assumed as a foun-
dation, and by making in this conception the successive changes
required by facts which it cannot explain.' And he goes on to
say that researches like his own, starting from general notions,
'can be useful in preventing this work from becoming hampered
by too narrow views, and progress of knowledge of the inter-
dependence of things from being checked by traditional
'This leads us into the domain of another science, that of
physics, into which the object of this work does not allow us to
go to-day.*
Riemann's work of 1854 put geometry in a new light. The
geometry he visions is non-Euclidean, not in the sense of
Lobatchewsky and Johann Bolyai, nor in that of Riemaiurs
own elaboration of the hypothesis of the obtuse angle (as
explained in chapter 16), but in a more comprehensive sense
depending on the conception of measurement. To isolate
measure-relations as the nerve of Riemann's theory is to do it an
injustice; the theory contains much more than & workable
philosophy of metrics, but this is one of its main features. No
paraphrase of Riemann's concise memoir can bring out all that
is hi it; nevertheless, we shall attempt to describe some of his
basic ideas, and we shall select three: the concept of a manifold,
the definition of distance, and the notion of curvature of a
A manifold is a class of objects (at least in common mathe-
matics) which is such that any member of the class can be
completely specified by assigning to it certain numbers, in &